Calculus Examples

$\cos (x) + \csc^{2} (x) + \csc (x) \cot (x)$

Step 1

Write $\cos (x) + \csc^{2} (x) + \csc (x) \cot (x)$ as a function.

$f (x) = \cos (x) + \csc^{2} (x) + \csc (x) \cot (x)$

Step 2

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 3

Set up the integral to solve.

$F (x) = \int \cos (x) + \csc^{2} (x) + \csc (x) \cot (x) d x$

Step 4

Split the single integral into multiple integrals.

$\int \cos (x) d x + \int \csc^{2} (x) d x + \int \csc (x) \cot (x) d x$

Step 5

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$\sin (x) + C + \int \csc^{2} (x) d x + \int \csc (x) \cot (x) d x$

Step 6

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$\sin (x) + C - \cot (x) + C + \int \csc (x) \cot (x) d x$

Step 7

Since the derivative of $- \csc (x)$ is $\csc (x) \cot (x)$ , the integral of $\csc (x) \cot (x)$ is $- \csc (x)$ .

$\sin (x) + C - \cot (x) + C - \csc (x) + C$

Step 8

Simplify.

$\sin (x) - \cot (x) - \csc (x) + C$

Step 9

The answer is the antiderivative of the function $f (x) = \cos (x) + \csc^{2} (x) + \csc (x) \cot (x)$ .

$F (x) =$ $\sin (x) - \cot (x) - \csc (x) + C$